On 9/19/2013 10:25 PM, Dan Christensen wrote:> On Thursday, September 19, 2013 11:07:23 PM UTC-4, fom wrote:>> On 9/19/2013 9:19 PM, Dan Christensen wrote:>>>>> On Thursday, September 19, 2013 7:36:04 PM UTC-4, fom wrote:>>>>>> On 9/19/2013 1:26 PM, Dan Christensen wrote:>>>>>>>>>>>>>>>>>>>>>>>>>>>> Why apart from? Why are you leaving it out?>>>>>>>>>>>>>>>>>>>>>>>>>> We can't divide by 0. Unless you want to assign a value to 0/0 as well.>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> And, what about division is "number theoretic"?>>>>>>>>>>>>>>>> As you can see in my proof, I am actually using the right-cancelability property of natural number multiplication:>>>>>>>>>> x*y = z*y & y=/=0 => x=z>>>>>>>>>> "Dividing" both sides by y, to cancel off the factor of y, OK?>>>>>>>>>>>>> Not really.>>>>>>>> Cancellability is a feature of group operations or, more>>>> generally magmas.>>>> It is also a feature (or theorem) of natural number arithmetic, e.g. x*2 = y*2 => x=y>>>>>>>> http://en.wikipedia.org/wiki/Cancellation_property>>>>>>>> So, again, you are invoking a property that is not strictly>>>> "number theoretic".>>>> I disagree.>>>>>>> The axiom of Peano's number theory that addresses equality>>>> between natural numbers (other than transitivity, symmetry,>>>> reflexiveness, and closure) is given by>>>>>>>> x=y <-> ( x+1 ) = ( y+1 )>>>>>>>> Wikipedia lists this as a conditional,>>>>>>>> ( x+1 ) = ( y+1 ) -> x=y>>>>>>>> It is axiom 8,>>>>>>>> http://en.wikipedia.org/wiki/Peano_axioms#The_axioms>>>>>>>> Now, since you continue to invoke "number theory", perhaps>>>> you could provide the formal theory which you seem to>>>> think that everyone knows and loves. It is clearly not>>>> Peano arithmetic.>> If you won't accept x*2 = y*2 => x=y, I afraid there is not much point in continuing this discussion.>

Once again, familiar discourse common to oneengaging in lies and evasions.

There has never been a point to this discussionbecause you have so badly mangled the mathematicsinvolved with the question.